UNSW MathSoc Championship Q7

Geometry Level 3

Let $f$ be the function defined by $f(x) = \begin{cases} x, \quad \text{if } x \geq 0\\ 0, \quad \text{otherwise.}\end{cases}$ Find the area of the region bounded by the $x$ -axis and the curve $y=f(x-1)-2f(x-5)+2f(x-7)-2f(x-9)+f(x-13)$ between $x=1$ and $x=13$ .

The answer is 28.

1 solution

Richard Xu
Mar 28, 2018

The region could be divided into 4 trapezoids (Note that the function is always linear), with their tops and bottoms at x=1, 5, 7, 9 ,13

Since f(1) = 0, f(5) = 4, f(7) = 2, f(9) = 4, f(13) =0, the area of the region is

S = (4+0)x(5-1)/2 + (4+2)x(7-5)/2 + (2+4)x(9-7)/2 + (4+0)x(13-9)/2 = 8 + 6 + 6 + 8 = 28

UNSW MathSoc Championship Q7

The answer is 28.

1 solution

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